Heat olive oil in a heavy, large pot over medium heat. Add the leeks and the garlic and stir. Add the potatoes and cook for 5 minutes, stirring often. Add the artichokes, stock, salt, and pepper and cook until the vegetables are tender, about 20 minutes.
Using a handheld immersion blender, or in a blender in batches*, puree the soup. Add the 2 tablespoons mascarpone and blend again to combine. In a small bowl, stir the remaining 1/3 cup mascarpone to soften.
Ladle the soup into serving bowls. Dollop the top of each of the soups with a spoonful of the softened mascarpone cheese and top the cheese with chives.
* When blending hot liquids: Remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.